Finite Group Databasebeta

← #29576281

Subgroups — #29576281

Order 15616.

724 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455815C211yesmaximal, characteristic
2C42×D244C211yesmaximal, characteristic, complemented
3#21444314C211yesmaximal, characteristic, complemented
4#21444243C211yesmaximal, characteristic, complemented
5#21443720C211yesmaximal, characteristic, complemented
6#21443652C211yesmaximal, characteristic, complemented
7C61×#1962C211yesmaximal, characteristic, complemented
8C2×C4×D244—21no—
9C2×C4×D244C2211yescharacteristic
10C2×C42×D61—21no—
11#16840073—21no—
12#16840039C412yescomplemented
13#16839954C412yescomplemented
14#16839895—21no—
15C4×#11758199—22no—
16#16839815C2211yescharacteristic
17#16839815C2211yescharacteristic
18C4×C4⋊Dic61—21no—
19C4×C4⋊Dic61C2211yescharacteristic
20C4×#11758198—22no—
21C4×#11758198—21no—
22#16839496—22no—
23#16839489—22no—
24C61×#532C2211yescharacteristic
25C61×#423C2211yescharacteristic
26C2×#104×C61—21no—
27C2×#104×C61—21no—
28C43×C61C2211yescharacteristic
29C22×D244D411yescharacteristic
30C22×C4×D61—22no—
31C22×C4×D61—21no—
32C2×#11758199—22no—
33C4×D244—28no—
34C4×D244—21no—
35C4×D244—21no—
36C4×D244C2×C412yes—
37C4×D244D412yes—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×#11758198—22no—
41C2×#11758198—21no—
42C2×C4×Dic61—22no—
43C2×C4×Dic61—21no—
44C42×D61—42no—
45C42×D61—42no—
46#14519710—22no—
47C4⋊#92418—21no—
48C4⋊#92418C2×C412yescomplemented
49#14519698—22no—
50#14519697—22no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53C4×#92418—21no—
54C4×#92418—21no—
55#14519684—22no—
56#14519684D412yes—
57C2×#41×C61—21no—
58C2×M16×C61—21no—
59C4⋊Q8×C61C2×C411yescharacteristic, complemented
60#127×C61C2×C411yescharacteristic, complemented
61#124×C61—21no—
62#121×C61—21no—
63#121×C61—21no—
64C4×D4×C61—21no—
65C4×D4×C61—21no—
66C2×C42×C61—21no—
67C2×C42×C61—21no—
68C2×C42×C61D411yescharacteristic
69C2×C42×C61D411yescharacteristic
70C2×C42×C61C2311yescharacteristic
71C4⋊C8×C61—21no—
72C4⋊C8×C61—21no—
73#104×C61—42no—
74#104×C61—42no—
75C23×D61—21no—
76#41×C61—41no—
77#41×C61—41no—
78#41×C61—41no—
79#41×C61—41no—
80C2×Q8×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61—41no—
85C22×C4×C61—21no—
86C22×C4×C61—21no—
87C22×C4×C61—41no—
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×Dic61—21no—
92C22×Dic61—22no—
93C2×D244—21no—
94C2×D244—24no—
95C2×D244C22⋊C412yes—
96C2×C4×D61—28no—
97C2×C4×D61—22no—
98C2×C4×D61—22no—
99C2×C4×D61—44no—
100C2×C4×D61—44no—
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×C4×D61—41no—
104C2×C4×D61—41no—
105M16×C61—41no—
106M16×C61—41no—
107C2×C8×C61—41no—
108C4⋊C4×C61—21no—
109C4⋊C4×C61—21no—
110C22⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112C42×C61—24no—
113C42×C61—21no—
114C42×C61—21no—
115C42×C61—21no—
116C42×C61—42no—
117C42×C61D812yescomplemented
118C42×C61C22×C411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61#4111yescharacteristic
122#11758199—44no—
123#11758198—28no—
124#11758198—22no—
125#11758198—22no—
126C4⋊Dic61—24no—
127C4⋊Dic61—21no—
128C4⋊Dic61C22⋊C412yes—
129C4×Dic61—44no—
130C4×Dic61—41no—
131C4×Dic61—41no—
132C2×#92418—22no—
133C2×#92418—41no—
134C23×C61—41no—
135C23×C61#12011yescharacteristic
136C22×D61—24no—
137C22×D61—22no—
138C22×D61—42no—
139C22×D61—41no—
140C22×D61—41no—
141Q8×C61—41no—
142Q8×C61—41no—
143D4×C61—42no—
144D4×C61—41no—
145D4×C61—41no—
146D4×C61—81no—
147D4×C61—81no—
148D4×C61—81no—
149C2×C4×C61—22no—
150C2×C4×C61—22no—
151C2×C4×C61—22no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—81no—
159C2×C4×C61—81no—
160C2×C4×C61C2×C22⋊C411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61C4×D411yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12311yescharacteristic
166C2×C4×C61C2×D811yescharacteristic
167C2×C4×C61#13511yescharacteristic
168C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
170C2×Dic61—24no—
171C2×Dic61—22no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—41no—
175C2×Dic61—41no—
176D244—24no—
177D244—22no—
178D244#10414yescomplemented
179C4×D61—416no—
180C4×D61—42no—
181C4×D61—42no—
182C4×D61—42no—
183C4×D61—42no—
184C8×C61—81no—
185#92418—42no—
186#92418—81no—
187#9233—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—81no—
193C22×C61#38611yescharacteristic
194C22×C61#44711yescharacteristic
195C22×C61#44911yescharacteristic
196C2×D61—24no—
197C2×D61—44no—
198C2×D61—44no—
199C2×D61—42no—
200C2×D61—42no—
201C4×C61—24no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—42no—
206C4×C61—81no—
207C4×C61C2×#10412yes—
208C4×C61C4×D811yescharacteristic
209C4×C61#44211yescharacteristic
210C4×C61#45911yescharacteristic
211C4×C61#46311yescharacteristic
212Dic61—44no—
213Dic61—42no—
214#3100—611no—
215C42×D4—611no—
216#1962D6111yescharacteristic, complemented
217#1502—611no—
218#1486—611no—
219#1303—611no—
220#1276—611no—
221C2×C61—21no—
222C2×C61—21no—
223C2×C61—161no—
224C2×C61#187911yescharacteristic
225C2×C61#189611yescharacteristic
226C2×C61#190611yescharacteristic
227D61—44no—
228#532C2×D6111yescharacteristic
229C2×C4×D4—611no—
230C2×C4×D4—1221no—
231C22×C42—1221no—
232#480—1221no—
233#474—612no—
234#459—612no—
235#447—1221no—
236C4×D8—1222no—
237#431—611no—
238#431—611no—
239#423C2×D6111yescharacteristic
240C2×#104—21no—
241C2×#104—21no—
242C4×C4⋊C4—611no—
243C4×C4⋊C4—1221no—
244C4×C22⋊C4—1222no—
245C4×C22⋊C4—1221no—
246C43C2×D6111yescharacteristic
247#327—1222no—
248D4⋊C8—1222no—
249C61#923311yescharacteristic, complemented, Sylow
250C2×#41—21no—
251C22×D4—611no—
252C23×C4—1222no—
253C23×C4—1221no—
254C2×D8—1222no—
255C2×M16—21no—
256C4⋊Q8C4×D6111yescharacteristic, complemented
257#127C4×D6111yescharacteristic, complemented
258#124—21no—
259#121—21no—
260#121—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4—612no—
264C4×D4—612no—
265C4×D4—1228no—
266C4×D4—1221no—
267C4×D4—1221no—
268C2×C4⋊C4—611no—
269C2×C4⋊C4—1221no—
270C2×C22⋊C4—1222no—
271C2×C22⋊C4—1221no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—1222no—
275C2×C42—1221no—
276C2×C42—2442no—
277C2×C42—2442no—
278C2×C42C22⋊D6111yescharacteristic
279C2×C42C22⋊D6111yescharacteristic
280C2×C42C22×D6111yescharacteristic
281#107—1222no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—612no—
285C4⋊C8—1221no—
286#104—42no—
287#104—42no—
288#102—1222no—
289#102—1222no—
290C22⋊C8—1221no—
291C22⋊C8—1221no—
292C4×C8—1221no—
293C4×C8—1221no—
294#95—612no—
295#95—1222no—
296C24—1221no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301C2×Q8—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—612no—
307C2×D4—1224no—
308C2×D4—1221no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—1228no—
313C22×C4—1222no—
314C22×C4—1221no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—2444no—
318C22×C4—2444no—
319C22×C4—2441no—
320C22×C4—2441no—
321C22×C4—2441no—
322C22×C4—2441no—
323C22×C4D4×D6111yescharacteristic
324C22×C4D4×D6111yescharacteristic
325C22×C4C2×C22⋊D6111yescharacteristic
326D8—2444no—
327M16—41no—
328M16—41no—
329C2×C8—41no—
330C2×C8—1222no—
331C2×C8—2441no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—612no—
335C4⋊C4—1224no—
336C4⋊C4—1221no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—1228no—
340C22⋊C4—1222no—
341C22⋊C4—1222no—
342C42—24no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—42no—
347C42—2444no—
348C42—2441no—
349C42—2441no—
350C42#1175819912yescomplemented
351C42C2×C4×D6111yescharacteristic
352C42#1175822211yescharacteristic
353C42D4×D6111yescharacteristic
354C42#1175822411yescharacteristic
355C23—41no—
356C23—1224no—
357C23—1222no—
358C23—2442no—
359C23—2441no—
360C23—2441no—
361C23#1451974511yescharacteristic
362Q8—41no—
363Q8—41no—
364D4—42no—
365D4—41no—
366D4—41no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—614no—
371D4—1224no—
372D4—1222no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—1224no—
385C2×C4—1222no—
386C2×C4—24416no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4—2442no—
395C2×C4C22⋊C4×D6111yescharacteristic
396C2×C4#1451973711yescharacteristic
397C2×C4C4×C22⋊D6111yescharacteristic
398C2×C4#1451974811yescharacteristic
399C2×C4#1451975011yescharacteristic
400C2×C4#1451975911yescharacteristic
401C2×C4C2×#1175819911yescharacteristic
402C2×C4#1451979711yescharacteristic
403C2×C4#1451980211yescharacteristic
404C2×C4#1451980311yescharacteristic
405C8—81no—
406C8—4881no—
407C8—2442no—
408C22—21no—
409C22—21no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—1224no—
414C22—2444no—
415C22—2444no—
416C22—2442no—
417C22—2442no—
418C22#1683963811yescharacteristic
419C22#1683989511yescharacteristic
420C22#1683990511yescharacteristic
421C4—24no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—42no—
426C4—81no—
427C4—2444no—
428C4—2442no—
429C4#104×D6112yes—
430C4C4×#1175819911yescharacteristic
431C4#1683987211yescharacteristic
432C4#1683995411yescharacteristic
433C4#1683997411yescharacteristic
434C2—21no—
435C2—21no—
436C2—161no—
437C2—2444no—
438C2#2144617011yescharacteristic
439C2#2144625911yescharacteristic
440C2#2144632311yescharacteristic
subgroupG/H
centerC2×C4#14519748
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1962C2
socleC22×C61#386
prime radical#21455815C2
cosocleC2×C42×C61C23
hypercenter#1962D61
hypocenterC61#9233
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1962

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C42×D244C2semidirect
3. #21444314C2semidirect
4. #21444243C2semidirect
5. #21443720C2semidirect
6. #21443652C2semidirect
7. C61×#1962C2semidirect
12. #16840039C4semidirect
13. #16839954C4semidirect
48. C4⋊#92418C2×C4semidirect
59. C4⋊Q8×C61C2×C4semidirect
60. #127×C61C2×C4semidirect
117. C42×C61D8semidirect
178. D244#104semidirect
216. #1962D61semidirect
249. C61#9233semidirect
256. C4⋊Q8C4×D61semidirect
257. #127C4×D61semidirect
350. C42#11758199semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#214558152
C42×D2442
#214443142
#214442432
#214437202
#214436522
C61×#19622
#923361